%%% figures.tex -- genera le figure XY-Pic per inclusione come .eps separati
%%%
%%% $Id: figures.tex,v 1.1.1.1 2002/01/10 10:01:51 ri Exp $
%%%
%%% This file generates a .dvi file from which the .eps figures are to
%%% be extracted:
%%%     dvips -Pamz -Pcmz -i -E figures.dvi -o fig.NUM
%%%
%auto-ignore
\documentclass[a4paper,fleqn,reqno]{amsart}

% questi servono per aggiustare:
\abovedisplayskip=0pt
\belowdisplayskip=0pt

\usepackage{color}
\newenvironment{eps}
  {\noindent{\color{white}\vrule width1pt height0pt depth1pt}\\[0pt]}
  {{\color{white}\hrule width1pt height1pt depth0pt}}


%%
%% Notazioni matematiche
%%
\usepackage[ps,dvips,arc,all,knot,frame,curve,poly]{xy}
\everyxy={/r24pt/:}

\usepackage{rg} % vertici e buche di grafi a nastro

%%
%% Notazioni matematiche di *questo* articolo
%%
\usepackage{miscmath} % simboli vari

\newcommand{\A}{\category{A}}     % generic category A
\newcommand{\B}{\category{B}}     % generic category B
\newcommand{\rev}{\sptext{rev}}
\newcommand{\catN}{\category{N}}     % generic category N
\newcommand{\Oo}[1][]{\operad{O}_{#1}}  % generic operad O
\newcommand{\RTD}[1][]{\operad{D}_{#1}} % operad of RT diagrams
\newcommand{\RT}[1][]{\operad{T}_{#1}}  % operad of RT graphs
\newcommand{\RG}[1][]{\operad{R}_{#1}}  % operad of Ribbon graphs
\newcommand{\SG}[1][]{\operad{G}_{#1}}  % operad of classical graphs
\newcommand{\freemc}[1]{#1^{\otimes}}        % free monoidal category 
\newcommand{\freemsc}[1]{#1^{\Box}}     % free monoidal signed category 

% operad dei grafi a nastro
\newcommand{\Ribbon}[1][]{\operad{R}_{#1}}

% decomposizione cellulare dei grafi a nastro
\newcommand{\Vertices}[1]{#1^{(0)}}
\newcommand{\Edges}[1]{#1^{(1)}}
\newcommand{\Holes}[1]{#1^{(2)}}

% medie sui grafi
\newcommand{\vv}[1]{V(#1)}
\newcommand{\biggint}[2][]{%
\Biggl\langle\Biggl\langle#2\Biggr\rangle\Biggr\rangle_{#1}}
\newcommand{\gint}[2][]{%
  \left\langle\!\left\langle#2\right\rangle\!\right\rangle_{#1}}
\newcommand{\ginttriv}[2][]{%
  \left\langle\!\left\langle#2\right\rangle\!\right\rangle_{#1}\triv}


%%
%% Cominciamo...
%%
\pagestyle{empty}
\begin{document}
\thispagestyle{empty}

%% fig-001: composition product of RT-graphs: rt.tex, gc.tex)
\begin{eps}
  \begin{equation*}
    {%
      {\xy%
        ,(0,-1.5);(0.6,-0.4)*+[F]{\ }**\crv{(0,-1.3)&(0.4,-0.9)%
          &(0.6,-0.4)}?(.5)*\dir{>}%
        ,(0.6,-0.4)*+[F]{\ };(0.6,0.3)*+[F]{\ }**\crv{(0.6,-0.2)}%
        ?(.75)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(0,1.5)**\crv{(0.5,0.5)&(0.4,0.7)}%
        ?(.85)*\dir{>}%
        ,(1.2,-1.5);(1.2,-1.1)*+[F]{\ }**\crv{(1.2,-1.3)&(1.2,-1.1)}%
        ?(.10)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(0.6,-0.4)*+[F]{\ }**\crv{%
          (0.8,-0.7)&(0.6,-0.5)}%
        ?(.5)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(1.2,1.5)**\crv{(0.7,0.5)&(0.8,0.7)}%
        ?(.5)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(2,1.5)**\crv{(1.2,-1.1)&(1.4,-0.9)&(1.5,-0.7)}%
        ?(.85)*\dir{<},%
        ,(0,-1.7)*{+},(1.2,-1.7)*{-}%
        ,(0,1.7)*{+},(1.2,1.7)*{-}%
        ,(2,1.7)*{-},%
        ,(-0.2,-1.5);(2.2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2.2,1.5)**\dir{-}%
        \endxy}\space\circ\space
      {\xy%
        ,(0.8,-1.5);(0.8,-0.5)*+[F]{\ }**\dir{-}%\crv{(0.8,-1.1)&(0.8,-0.7)&}%
        ?(.4)*\dir{<}%
        ,(0.8,-0.5)*+[F]{\ };(0,1.5)**\crv{(0.4,-0.1)&(0.2,0.3)}%
        ?(.85)*\dir{>}%
        ,(0.8,-0.5)*+[F]{\ };(1.2,1.5)**\crv{(0.9,-0.4)&%
          (1.2,-0.3)&(1.6,0.3)&(1.6,0.7)}%
        ?(.85)*\dir{<}%
        ,(0.8,-1.7)*{-},(0,1.7)*{+}%
        ,(1.2,1.7)*{-}%
        ,(-0.2,-1.5);(2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2,1.5)**\dir{-}%
        \endxy}
      \quad = \quad
      {\xy%
        ,(0.8,-1.5);(0.8,-1)*+[F]{\ }**\dir{-}%
        ?(.25)*\dir{<}%
        ,(0.8,-1)*+[F]{\ };(0,0)**\crv{(0.4,-0.8)&(0.2,-0.6)}%
        ?(.85)*\dir{>}%
        ,(0.8,-1)*+[F]{\ };(1.2,0.2)*+[F]{\ }**\crv{(0.9,-0.95)&%
          (1.2,-0.9)&(1.6,-0.6)&(1.6,-0.4)&(1.2,0.0)}%
        ?(.35)*\dir{<}%
        ,(0,0);(0.6,0.6)*+[F]{\ }**\crv{(0,0.2)&(0.4,0.3)%
          &(0.6,0.6)}%
        ,(0.6,0.6)*+[F]{\ };(0.6,1.1)*+[F]{\ }**\dir{-}%
        ?(.75)*\dir{>}%
        ,(0.6,1.1)*+[F]{\ };(0,1.5)**\crv{(0.5,1.1)&(0.4,1.3)}%
        ?(.85)*\dir{>}%
        ,(1.2,0.2)*+[F]{\ };(0.6,0.6)*+[F]{\ }**\crv{%
          (0.8,0.3)&(0.6,0.7)}%
        ?(.5)*\dir{>}%
        ,(0.6,1.1)*+[F]{\ };(1.2,1.5)**\crv{(0.7,1.2)&(0.8,1.3)}%
        ?(.5)*\dir{<}%
        ,(1.2,0.2)*+[F]{\ };(2,1.5)**\crv{(1.2,0.2)&(1.4,0.3)&(1.5,0.4)}%
        ?(.85)*\dir{<},%
        ,(-0.2,0);(2.2,0)**\dir{.}%
        ,(0,1.7)*{+},(1.2,1.7)*{-}%  
        ,(2,1.7)*{-},%
        ,(-0.2,1.5);(2.2,1.5)**\dir{-}%
        ,(0.8,-1.7)*{-}%
        ,(-0.2,-1.5);(2,-1.5)**\dir{-}%
        \endxy}
      }
  \end{equation*}
\end{eps}
\clearpage

%% fig-002: tensor product of RT-graphs: rt.tex, gc.tex
\begin{eps}
  \begin{equation*}
    {%
      {\xy%
        ,(0,-1.5);(0.6,-0.4)*+[F]{\ }**\crv{(0,-1.3)&(0.4,-0.9)%
          &(0.6,-0.4)}?(.5)*\dir{>}%
        ,(0.6,-0.4)*+[F]{\ };(0.6,0.3)*+[F]{\ }**\crv{(0.6,-0.2)}%
        ?(.75)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(0,1.5)**\crv{(0.5,0.5)&(0.4,0.7)}%
        ?(.85)*\dir{>}%
        ,(1.2,-1.5);(1.2,-1.1)*+[F]{\ }**\crv{(1.2,-1.3)&(1.2,-1.1)}%
        ?(.10)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(0.6,-0.4)*+[F]{\ }**\crv{%
          (0.8,-0.7)&(0.6,-0.5)}%
        ?(.5)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(1.2,1.5)**\crv{(0.7,0.5)&(0.8,0.7)}%
        ?(.5)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(2,1.5)**\crv{(1.2,-1.1)&(1.4,-0.9)&(1.5,-0.7)}%
        ?(.85)*\dir{<},%
        ,(0,-1.7)*{+},(1.2,-1.7)*{-}%
        ,(0,1.7)*{+},(1.2,1.7)*{-}%
        ,(2,1.7)*{-},%
        ,(-0.2,-1.5);(2.2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2.2,1.5)**\dir{-}%
        \endxy}\space\otimes\space
      {\xy%
        ,(0.8,-1.5);(0.8,-0.5)*+[F]{\ }**\dir{-}%\crv{(0.8,-1.1)&(0.8,-0.7)&}%
        ?(.4)*\dir{<}%
        ,(0.8,-0.5)*+[F]{\ };(0,1.5)**\crv{(0.4,-0.1)&(0.2,0.3)}%
        ?(.85)*\dir{>}%
        ,(0.8,-0.5)*+[F]{\ };(1.2,1.5)**\crv{(0.9,-0.4)&%
          (1.2,-0.3)&(1.6,0.3)&(1.6,0.7)}%
        ?(.85)*\dir{<}%
        ,(0.8,-1.7)*{-},(0,1.7)*{+}%
        ,(1.2,1.7)*{-}%
        ,(-0.2,-1.5);(2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2,1.5)**\dir{-}%
        \endxy}
      \quad = \quad
      {\xy%
        ,(0,-1.5);(0.6,-0.4)*+[F]{\ }**\crv{(0,-1.3)&(0.4,-0.9)%
          &(0.6,-0.4)}?(.5)*\dir{>}%
        ,(0.6,-0.4)*+[F]{\ };(0.6,0.3)*+[F]{\ }**\crv{(0.6,-0.2)}%
        ?(.75)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(0,1.5)**\crv{(0.5,0.5)&(0.4,0.7)}%
        ?(.85)*\dir{>}%
        ,(1.2,-1.5);(1.2,-1.1)*+[F]{\ }**\crv{(1.2,-1.3)&(1.2,-1.1)}%
        ?(.10)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(0.6,-0.4)*+[F]{\ }**\crv{%
          (0.8,-0.7)&(0.6,-0.5)}%
        ?(.5)*\dir{>}%
        ,(0.6,0.3)*+[F]{\ };(1.2,1.5)**\crv{(0.7,0.5)&(0.8,0.7)}%
        ?(.5)*\dir{<}%
        ,(1.2,-1.1)*+[F]{\ };(2,1.5)**\crv{(1.2,-1.1)&(1.4,-0.9)&(1.5,-0.7)}%
        ?(.85)*\dir{<},%
        ,(0,-1.7)*{+},(1.2,-1.7)*{-}%
        ,(0,1.7)*{+},(1.2,1.7)*{-}%
        ,(2,1.7)*{-},%
        ,(-0.2,-1.5);(2.2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2.2,1.5)**\dir{-}%
        \endxy}
      {\xy%
        ,(0.8,-1.5);(0.8,-0.5)*+[F]{\ }**\dir{-}%\crv{(0.8,-1.1)&(0.8,-0.7)&}%
        ?(.4)*\dir{<}%
        ,(0.8,-0.5)*+[F]{\ };(0,1.5)**\crv{(0.4,-0.1)&(0.2,0.3)}%
        ?(.85)*\dir{>}%
        ,(0.8,-0.5)*+[F]{\ };(1.2,1.5)**\crv{(0.9,-0.4)&%
          (1.2,-0.3)&(1.6,0.3)&(1.6,0.7)}%
        ?(.85)*\dir{<}%
        ,(0.8,-1.7)*{-},(0,1.7)*{+}%
        ,(1.2,1.7)*{-}%
        ,(-0.2,-1.5);(2,-1.5)**\dir{-}%
        ,(-0.2,1.5);(2,1.5)**\dir{-}%
        \endxy}
      }
  \end{equation*}
\end{eps}
\clearpage

%% fig-003: RRT moves: rt.tex, gc.tex
\begin{eps}
    {%
      \begin{tabular}{cc}
        \(
        {\xy(-.5,-1);(.5,1)**\crv{(-.5,.5)&(-.2,0.5)&(.2,-0.5)&(.5,-.5)}\endxy}
        \quad\leftrightarrow\quad
        {\xy(0,-1);(0,1)**\dir{-}\endxy}
        \quad\leftrightarrow\quad
        {\xy(.5,-1);(-.5,1)**\crv{(.5,.5)&(.2,0.5)&(-.2,-0.5)&(-.5,-.5)}\endxy}
        \)
        &
        \(
        {\xy
          \hunder~{(0,.7)}{(.5,.7)}{(0,0.2)}{(.5,.2)}
          \hcap~{(0.5,.7)}{(1,.7)}{(0.5,0.2)}{(1,0.2)}
          \hover~{(0,-.2)}{(.5,-.2)}{(0,-.7)}{(.5,-.7)}
          \hcap~{(0.5,-.2)}{(1,-.2)}{(0.5,-.7)}{(1,-.7)}
          ,(0,.7);(0,1)**\dir{-},(0,.2);(0,-.2)**\dir{-}%
          ,(0,-.7);(0,-1)**\dir{-}
          \endxy}
        \quad\leftrightarrow\quad
        {\xy
          (0,-1);(0,1)**\dir{-},
          \ar@[grey] (0,-1.00);(+0.50,-1.00),
          \ar@[grey] (0,-0.75);(+0.25,-0.75),
          \ar@{-} (0,-0.50)*[grey]{\bullet};(0,-0.50),
          \ar@[grey] (0,-0.25);(-0.25,-0.25),
          \ar@[grey] (0,+0.00);(-0.50,+0.00),
          \ar@[grey] (0,+0.25);(-0.25,+0.25),
          \ar@{-} (0,+0.50)*[grey]{\bullet};(0,-0.50),
          \ar@[grey] (0,+0.75);(+0.25,+0.75),
          \ar@[grey] (0,+1.00);(+0.50,+1.00),
          \endxy}
        \)
        \\
        (RRT1)
        &
        (RRT2)
        \\[2ex]
        \(
        {\xy
          \vcrossneg~{(0,1.5)}{(.5,1.5)}{(0,0.5)}{(.5,0.5)}
          \vcrossneg~{(.5,0.5)}{(1,0.5)}{(.5,-.5)}{(1,-.5)}
          \vcrossneg~{(0,-0.5)}{(.5,-0.5)}{(0,-1.5)}{(.5,-1.5)}
          ,(1,1.5);(1,.5)**\dir{-},(0,.5);(0,-.5)**\dir{-},
          (1,-.5);(1,-1.5)**\dir{-}
          \endxy}
        \quad\leftrightarrow\quad
        {\xy
          \vcrossneg~{(0.5,1.5)}{(1,1.5)}{(0.5,0.5)}{(1,0.5)}
          \vcrossneg~{(0,0.5)}{(.5,0.5)}{(0,-.5)}{(.5,-.5)}
          \vcrossneg~{(0.5,-0.5)}{(1,-0.5)}{(0.5,-1.5)}{(1,-1.5)}
          ,(0,1.5);(0,.5)**\dir{-},(1,.5);(1,-.5)**\dir{-},
          (0,-.5);(0,-1.5)**\dir{-}
          \endxy}
        \)
        &
        \(
        {\xy*!LC\xybox{\vcross\vcrossneg}\endxy}
        \quad\leftrightarrow\quad
        {\xy(-.2,-1);(-.2,1)**\dir{-},(.2,-1);(.2,1)**\dir{-}\endxy}
        \quad\leftrightarrow\quad
        {\xy*!LC\xybox{\vcrossneg\vcross}\endxy}
        \)
        \\
        (RRT3)
        &
        (RRT4)
        \\[2ex]
        \({\xy(0.6,0):(0,0.75)::
          \vcrossneg~{(0,1.5)}{(.5,1.5)}{(0,1)}{(.5,1)}
          \vcap~{(0.5,2)}{(1,2)}{(0.5,1.5)}{(1,1.5)}
          \vcap~{(-0.5,0.5)}{(0,0.5)}{(-0.5,1)}{(0,1)}
          \vcap~{(0,2.5)}{(1.5,2.5)}{(0,1.5)}{(1.5,1.5)}
          \vcap~{(-1,0.1)}{(0.5,0.1)}{(-1,1)}{(0.5,1)}
          \vcrossneg~{(0,-1.5)}{(.5,-1.5)}{(0,-1)}{(.5,-1)}
          \vcap~{(0.5,-2)}{(1,-2)}{(0.5,-1.5)}{(1,-1.5)}  
          \vcap~{(-0.5,-.5)}{(0,-.5)}{(-0.5,-1)}{(0,-1)}
          \vcap~{(0,-2.5)}{(1.5,-2.5)}{(0,-1.5)}{(1.5,-1.5)}     
          \vcap~{(-1,-0.1)}{(0.5,-0.1)}{(-1,-1)}{(0.5,-1)},
          (-1,1);(-1,2.5)**\dir{-},(1,1.5);(1,0)**\dir{-},
          (1.5,1.5);(1.5,0)**\dir{-},
          (-1,-1);(-1,-2.5)**\dir{-},(1,-1.5);(1,0)**\dir{-},
          (1.5,-1.5);(1.5,0)**\dir{-},
          (-.5,1);(-.5,2.5)**\dir{-},(-.5,-1);(-.5,-2.5)**\dir{-}
          \endxy}
        \quad\leftrightarrow\quad
        {\xy(0.6,0):(0,0.75)::
          (-0.2,2.5);(-0.2,-2.5)**\dir{-},
          (0.2,2.5);(0.2,-2.5)**\dir{-}\endxy}
        \quad\leftrightarrow\quad
        {\xy(0.6,0):(0,0.75)::
          \vcross~{(0,1.5)}{(.5,1.5)}{(0,1)}{(.5,1)}
          \vcap~{(0.5,2)}{(1,2)}{(0.5,1.5)}{(1,1.5)}
          \vcap~{(-0.5,0.5)}{(0,0.5)}{(-0.5,1)}{(0,1)}
          \vcap~{(0,2.5)}{(1.5,2.5)}{(0,1.5)}{(1.5,1.5)}
          \vcap~{(-1,0.1)}{(0.5,0.1)}{(-1,1)}{(0.5,1)}
          \vcross~{(0,-1.5)}{(.5,-1.5)}{(0,-1)}{(.5,-1)}
          \vcap~{(0.5,-2)}{(1,-2)}{(0.5,-1.5)}{(1,-1.5)}  
          \vcap~{(-0.5,-.5)}{(0,-.5)}{(-0.5,-1)}{(0,-1)}
          \vcap~{(0,-2.5)}{(1.5,-2.5)}{(0,-1.5)}{(1.5,-1.5)}     
          \vcap~{(-1,-0.1)}{(0.5,-0.1)}{(-1,-1)}{(0.5,-1)},
          (-1,1);(-1,2.5)**\dir{-},(1,1.5);(1,0)**\dir{-},
          (1.5,1.5);(1.5,0)**\dir{-},
          (-1,-1);(-1,-2.5)**\dir{-},(1,-1.5);(1,0)**\dir{-},
          (1.5,-1.5);(1.5,0)**\dir{-},
          (-.5,1);(-.5,2.5)**\dir{-},(-.5,-1);(-.5,-2.5)**\dir{-}
          \endxy}
        \)
        &
        \(
        {\xy*!LC\xybox{
            \vcap~{(-0.5,1)}{(0,1)}{(-0.5,0.5)}{(0,0.5)}
            \vcap~{(-0.7,1.2)}{(0.2,1.2)}{(-0.7,0.5)}{(0.2,0.5)}
            \vcap~{(-0.9,1.4)}{(0.4,1.4)}{(-0.9,0.5)}{(0.4,0.5)}
            \vcap~{(-1.1,1.6)}{(0.6,1.6)}{(-1.1,0.5)}{(0.6,0.5)}
            \vcap~{(0,-.5)}{(0.6,-.5)}{(0,0)}{(0.6,0)}  
            \vcap~{(0.2,-.3)}{(0.4,-.3)}{(0.2,0)}{(0.4,0)}  
            \vcap~{(-0.5,-.9)}{(0.8,-.9)}{(-0.5,0)}{(0.8,0)}  
            \vcap~{(-0.7,-1.1)}{(1,-1.1)}{(-0.7,0)}{(1,0)}
            ,(0.1,0.25)*+[F]{\ };(0,0.5)**\crv{(0,0.3)}
            ,(0.1,0.25)*+[F]{\ };(0.2,0.5)**\crv{(0.2,0.3)}
            ,(0.1,0.25)*+[F]{\ };(0,0)**\crv{(0,0.2)}
            ,(0.1,0.25)*+[F]{\ };(0.2,0)**\crv{(0.2,0.2)}
            ,(0.8,0);(0.2,1.5)**\crv{(0.8,1)&(0.2,1.2)}
            ,(1,0);(1.2,1.5)**\crv{(1,0.5)&(1.2,1)}
            ,(0.7,1,4)*{\cdots}
            ,(-0.9,0);(-0.3,-1.5)**\crv{(-0.9,-1)&(-0.3,-1.2)}
            ,(-1.1,0);(-1.3,-1.5)**\crv{(-1.1,-0.5)&(-1.3,-1)}
            ,(-0.8,-1,4)*{\cdots}
            ,(0.4,0.5);(0.4,0)**\dir{-}
            ,(0.6,0.5);(0.6,0)**\dir{-}
            ,(-0.5,0.5);(-0.5,0)**\dir{-}
            ,(-0.7,0.5);(-0.7,0)**\dir{-}
            ,(-0.9,0.5);(-0.9,0)**\dir{-}
            ,(-1.1,0.5);(-1.1,0)**\dir{-}
            }
          \endxy}
        \quad\leftrightarrow\quad
        {\xy*!LC\xybox{
            ,(0,0)*+[F]{\ };(-0.5,-1.5)**\crv{(-0.5,-1.1)}
            ,(0,0)*+[F]{\ };(0.5,-1.5)**\crv{(0.5,-1.1)}
            ,(0,0)*+[F]{\ };(-0.5,1.5)**\crv{(-0.5,1.1)}
            ,(0,0)*+[F]{\ };(0.5,1.5)**\crv{(0.5,1.1)}
            ,(0,1.4)*{\cdots},(0,-1.4)*{\cdots}
            }
          \endxy}
        \)
        \\
        (RRT5)
        &
        (RRT6)
        \\[2ex]
        \(
        {\xy
          *!LC\xybox{(0.66,0):(0,1.25)::
            (0,1)*+[F]{\ };% 
            (-1,0)**\dir{-},(-0.5,0)**\dir{-},%
            (0,0.5)*+{\ldots},(1,0)**\dir{-},%
            (-.4,1.4)**\dir{-},(-0.2,1.4)**\dir{-},%
            (0,1.5)*+{\ldots},(0.45,1.45)**\dir{-},%
            ,(-1.5,0);(1.5,2)**\crv{(-1.5,1.15)&(-1.5,1.5)%
              &(1.5,1.5)&(1.5,1.85)}
            ,(-.6,1.6);(-1,2)**\dir{-},(-0.3,1.6);(-0.5,2)**\dir{-}
            ,(0.65,1.65);(1,2)**\dir{-}
            }
          \endxy}
        \quad\leftrightarrow\quad
        {\xy
          *!LC\xybox{(0.66,0):(0,1.25)::
            (0,-1)*+[F]{\ };% 
            (1,0)**\dir{-},(0.5,0)**\dir{-},%
            (0,-0.5)*+{\ldots},(-1,0)**\dir{-},%
            (.4,-1.4)**\dir{-},(0.2,-1.4)**\dir{-},%
            (0,-1.5)*+{\ldots},(-0.45,-1.45)**\dir{-},%
            ,(1.5,0);(-1.5,-2)**\crv{(1.5,-1.15)&(1.5,-1.5)%
              &(-1.5,-1.5)&(-1.5,-1.85)}
            ,(.6,-1.6);(1,-2)**\dir{-},(0.3,-1.6);(0.5,-2)**\dir{-}
            ,(-0.65,-1.65);(-1,-2)**\dir{-}
            }
          \endxy}
        \)
        &
        \( 
        {\xy
          *!LC\xybox{(0.66,0):(0,1.25)::
            (0,1)*+[F]{\ };% 
            (-1,0)**\dir{-},(-0.5,0)**\dir{-},%
            (0,0.5)*+{\ldots},(1,0)**\dir{-},%
            (-1,2)**\dir{-},(-0.5,2)**\dir{-},%
            (0,1.5)*+{\ldots},(1,2)**\dir{-},%
            ,(-1.5,0);(-.5,1.4)**\crv{(-1.5,1.15)&(-1.5,1.4)}
            ,(0.6,1.5);(1.5,2)**\crv{(1.5,1.7)&(1.5,2)}
            }
          \endxy}
        \quad\leftrightarrow\quad
        {\xy
          *!LC\xybox{(0.66,0):(0,1.25)::
            (0,-1)*+[F]{\ };% 
            (1,0)**\dir{-},(0.5,0)**\dir{-},%
            (0,-0.5)*+{\ldots},(-1,0)**\dir{-},%
            (1,-2)**\dir{-},(0.5,-2)**\dir{-},%
            (0,-1.5)*+{\ldots},(-1,-2)**\dir{-},%
            ,(1.5,0);(.5,-1.4)**\crv{(1.5,-1.15)&(1.5,-1.4)}
            ,(-0.6,-1.5);(-1.5,-2)**\crv{(-1.5,-1.7)&(-1.5,-2)}
            }
          \endxy}
        \)
        \\
        (RRT7)
        &
        (RRT8)
      \end{tabular}
      }
\end{eps}
\clearpage

\end{document}%

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